Microsoft Word Copula vmt 6March09. doc



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T Copula VMT 6March09
MODELING METHODOLOGY
In this section, we present the structure of the copula-based joint multinomial logit – regression modeling framework to jointly model vehicle type choice and usage. First, the structure of the vehicle type choice model component is discussed, then the vehicle mileage model component is presented, and finally the joint structure between these two model components is described. The procedure used for model estimation is also presented in this section.
The Vehicle Type Choice Model Component
Let
( = 1, 2, ..., )
q q
Q
and
(
1, 2, ... )
i i
I
=
be the indices to represent households and vehicle types, respectively. The vehicle type choice model component takes the familiar discrete choice formulation. Consider the following equation that represents the utility structure of the vehicle type choice model
*
'
qi
i qi
qi
u
x
β
ε
=
+
(1) In the equation above,
*
qi
u
is the latent utility that the
th
q
household derives from acquiring a vehicle of type
i
,
qi
x
is a column vector of household attributes (including a constant, demographics, and activity- travel environment characteristics) affecting the utility,
i
β
is the corresponding coefficient
(column)vector, and
qi
ε
is the error term capturing the effects of unobserved factors on the utility associated with vehicle type
i
. With this utility specification, as with any discrete choice model, a household (q) is assumed to choose a vehicle of type
i
if it is associated with the maximum utility among all
I
vehicle types that is, if
1, 2,..., ,
*
*
max
j
I j i
qi
qj
u
u
=

>
(2) Next, following Lee (1983), the polychotomous discrete choice model is recast in the form of a series of binary choice formulations, one for each vehicle type. To do so, let
qi
R be a dichotomous variable that takes the values 0 and 1, with
1
qi
R
=
if the
th
i
alternative is chosen by the
th
q
household and
0
qi
R
=
otherwise. Subsequently, substituting
'
i qi
qi
x
β
ε
+
for
*
qi
u
[from Equation (1)] in Equation
(2), one can represent the discrete choice model formulation equivalently as
,
1 if, 2, ... )
'
qi
i qi
qi
R
x
i
I
β
ν
=
>
=
(3)
1, 2,..., ,
*
where
{
max
}
j
I j i
qi
qj
qi
u
ν
ε
=

=

(4) Equation (3) represents a series of binary choice formulations, which is equivalent to the multinomial discrete choice model of vehicle type. In this equation, the distribution of the
qi
ν
term depends on the distributional assumptions of the
qi
ε
terms see Equation (4)]. The distribution of the
qi
ν
terms, in turn, will determine the form of vehicle type choice probability expressions. For example, type extreme value distributed
qi
ε
terms that are independent (across
i
) and identically distributed imply a logistic distribution for the
qi
ν
terms, and, consequently, the vehicle type choice probability expressions resemble the multinomial logit probabilities.



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